(Note: I asked a very similar question, only after which I realized that I had asked something other than what I had in mind, so I'm posting what I truly intended to ask here.)
I am interested in density of $V \cap W$ in $W$ for a closed subspace $W$ of a completion of a pre-Hilbert space $V$. Does there exist $W$ for which $V \cap W$ is not dense in $W$? If so, what are $W$ for which $V\cap W$ is dense in $W$?